Assessment of errors in approximate solutions of differential equations

T H E

TECHNISCHE HOGESCHOOL

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TECHNISCHE ÜNIVgRSITEIT D E L F T ^ ^ O R T J T O J ^ J ^ LüCHTVAAfIT-r ^^ITTESHHfEK^ ^ , , , _

&, - .December, 13h.i

Kluyverweg 1 - 2629 HS DELR

12 Juli 1950

C R A N P I E L D

Assessment of Errors in Approximate Solutions of Differential Equations.

by

-¥. J. Duncan, D.Sc., P.R.S., Professor of Aerodynamics at the College of Aeronautics, Cranfield.

^^OoO . .

SUMMARY

-The term assessment is applied to any process which enables us to set rigid bounds to the error or to estimate its value. It is shown that upper and lower bounds can be

assigned whenever the Green' s function of the problem is one-signed; this is true in many important problems. Another method is applicable to step by step solutions of ordinary

differential equations, linear or non-linear, and depends on use of the "index" of the process of integration. Lastly, ;bhe error in a linear problem can be estimated when an

approximation to the Grepn' s function is known.

CONTENTS

Introduction: Meaning of Assessment

Bounds to the Errors of Problems having One-Signed Green's Functions.

2.1. Outline of the Method.

2.2. Errors in the Basic Solution, 2.3. Cases where the Boundary

Conditions are not Homogeneous, 2.4. An Important Case where the

Green's Function is One-Signed.

A Method for Estimating the Errors in Step by Step Solutions of Sets of Ordinary Differential Equations,

3.1. Outline of the Method. 3.2. Extension of the Method to

Partial Differential Equations.

Estimation of the Errors in Approximate Solutions of Linear Problems when an Approximation to the Green's Function is kno\7n.

- 3

1. INTRODUCTION; MEAI-IING OP ASSESSMENT,

The word assessment is used to cover any process which enables us to estimate or delimit. There are two principal kinds of assessments of error:

(a) the fixing of rigid upper and lower bounds to the error.

(b) the estimation, more or less closely, of the error.

ll/hen it is possible to fix rigid bounds to the error these are

usually rather T/idely separated and the assessment is correspondingly crude, but when the bounds are both rigid and close vre have a most useful assessment. However, when we have not discovered how to set rigid bounds, or when these are insufficiently close, we must have recourse to some method of estimation. Such an estimation may yield results of ample accuracy for the purposes of applied mathematics even when lacking ideal precision. It is to be remarked that we exclude strict evaluation of the error from

consideration since, if this could be done, we should have a method of exact solution. An essential requirement is that any method of assessment shall be applicable throughout the whole region of

integration,

We shall here confine attention to differential equations with bo\indary conditions v/hich render the solution unique. No attempt at a general treatment will be made, but the following items

vrxll be discussed:

(1) Bounds to the errors of ordinary and partial differential problems having one-signed Green's functions.

(2) Estimation of the errors of the step by step solutions of ordinary differential equations, or sets of these, -vvlth one point boundary conditions.

(5) Estimation of the errors of linear problems when an approximation to the Green's function

is known,

Item (l) is concerned only with linear problems and may appear of very limited applicability; in fact it covers some very important problems such as Poisson' s equation with fixed boundary values of the unkno-'.-m, Item (2) covers non-linear problems,

2. BOmmS TO THE ERRORS OF FROELEi>/[S HAVING ONE-SIGI^IED GREEN'S FTOTCTIONS. 2,1. Outline of the Method,

lie consider an ordinary or partial linear differential equation with boundary conditions of the linear and homogeneous type

which render the solution unique. We suppose further that the Green' s function of the problem is one-signed.

Let the differential equation be

Z ^ c/^= f .... (2.1,1)

•where i-A is a linear ordinary or partial differential operator,

fp

is the unknown and f is a given function of the independent

variable or variables. Also let

A O- = 1 .... (2.1,2)

with the same boundary conditions as for 4-' , We shall call O"

the basic solution, and it is one-signed on account of our assumption

about thi/ Green's function. Suppose that

(u ^

is some approximation

to

<'p

which exactly satisfies the boundary conditions and let

Z\<^a

- f = f ,

.... (2.1,3)

so

^

is the residual in the differential equation corresponding

to <p., and is, in. general, a function of the independent v-oriables.

Let

£ = absolute maximum value of £ in

•^

the region of integration

and ^ = absolute minimum value of (; in

2 the region of integration.

Then (

f

~ ^i) is everywhere negative or zero and (^ -'^2)

everywhere positive or zero in the region.

For definiteness let the Green's function be everywhere

positive in the region. Then by equations (2.1,1) - (2.1,3)

^ ( f - f a -^ ( 1 Ö- ) = ^ 1 -

( ^•

0

and consequently

f - f a ^

or "P > 4^a "

Also / \ ( 4^ - Cp +

fi

-f

10-.

f2 ^

= ^ 2 - ^ - °

and ^^ =?c

<p^ - € 2 ^ .

Finally

f a - ^1 ^ ^ f ^ "Pa- ^2 ^- ....(2.1,i..)

/If

5

-If the Green's function were everywhere negative we should have

- f e f ^ . C^cr. ....(2.1,5)

These inequalities call for two

remarks:-(a) Provided that v" and C Q are small, a rough approximation to Q" will usually give

sufficient information about the errors in *^, (b) A large but highly localised error in cP

^ a or its derivatives, will yield numerically large values for one or both of £ ^, £ and will greatly widen the bounds in the inequalities. It is therefore important to avoid such large local errors when the present method of assessment is used. 2,2. Errors in the Basic Solution,

Let CT be an approximation to Ct and

/>, A C T 1 =Y-| )

(2.2,1)

Also l e t YT J •y\ be the absolute maximum and absolute minimum v a l u e s , r e s p e c t i v e l y , of r i . Then

A ( (J - O-^ + -n i f 5 ' ) = ^ i - ' Y ) ^ 0

and A, C c r - CT, + Tl o (T ) =

_{a T 2 C r ^ = ->^2}

Hence if the Green's function is positive

O', _ cr^

1

*c

a" ^

'h

'•^T2

but i f the Green's function i s neg;ative

a • • 9 \ ^e i- f ^J

cr,

01

t' 2 / l

2 . 3 . Oases where the Boundary Conditions are Not Homogeneous,

When the boundary c o n d i t i o n s are l i n e a r but not homogeneous we can reduce the problem t o one \7ith homogeneous conditions as

follows. Let /3 be a convenient function which s a t i s f i e s the boundary - c o n d i t i o n s . Then

>= f-^

satisfies linear and homogeneous boundary conditions and

^ -^= f - A ƒ3 .... (2.3,2)

•vdiere the function on the right hand side of the equation is known,

2,1+, An Important Case where the Green' s Function is One-Signed.

By way of example we shall show that the Green's function for Poisson's equation is negative, the region of integration being the space interior to a closed svirface B upon which the solution is to vanish,

Let the differential equation to be solved be

Vc^)=

\b

.... (2.4,1)

where \l/ is everyshere positive within B and Cp is sero on B . Then either C^ is everywhere negative within B or it is positive at some point P within B. If it is positive at P it must be also positive everywhere within a closed surface C surrounding P and vanish on C ; moreover C is entirely m t h i n B or coincides Vidth it vvholly or partly and ^p is therefore everywhere positive within C , Apply Green's theorem to the space within G ,

A 2

V J

ihÉ] + cp \J cp I dxdydz

= 0 , . . . (2.i^,2)

where iL-L is the rate of change of <f^ along the inward drawn

è y

noimal to C and d ^ is the element of the surface of C . But according to our hypothesis the integrand is ev-..:rywhere positive

within C and the integral cannot vanish. Accordingly ^ cannot be positive anywhere within B , and, since '^j^ is an arbitrary

positive function, it follows that the Green's function is always negative. The same theorem is true in two dimensions.

This proposition was applied by the writer some years ago to delimit the errors in approximate solutions of special problems in the theory of elasticity 1,2, in place of the basic function O'

the Prandtl torsional stress function "^J-? was used; this satisfies the two-dimensional equation;

\ 7 ^

+ 2 = 0 ....

and vanishes on the closed boundary, so

it' = - 2 o- .... (2.if,4)

It was shown to be possible to place very close bounds to the errors in approximations to ^Jj itself l»^ and to a stress function arising in the St, Venant theory of flexure ^.

- 7

3. A METHOD FOR ESTIMTING THE ERRORS IN STEP BY STEP SOLUTIONS OF SETS OF ORDINARY DIFFERENTIJU:. EQUATIONS.

3.1. Outline of the Method.

Briefly, the method is based on the idea of extrapolation towards the lim.it of the step by step solution corresponding to a vanishingly small interval. Suppose that the independent variable is _t , the range of integration a to b , and n the number of equal intervals used in the step by step process. Then, provided that the process is completely regular and the boundary values of the unknowns ejre all given for a (say), we may assume that the error in the value of the dependent variable Xp can be expanded in the series

i^ (t) := n"'' ( pQ + Ci n"^ + ^2 ^'^ + ®*°- ) •••• (^•^'^) provided also that n is not too small. The number k is

characteristic of the particular process used and is called its index. Now when the interval is sufficiently small we may as a first

approximation retain only the first or dominant term in the expansion and \wite

^^(t) =

e^ n'^.

,,., (3.1,2)

Since k will be knovm it becomes possible to calculate '^^ , when the approximate solution has been obtained for two values of n , Thus we shall have

x^(t) = x^i(t) + 6o njk = x^2^t) + êo rq

where x •,(t), x p(t) are the approximations to x^(t) with n^ and np intervals respectively. The last equations yield

G = !k2l!LLf£ii^ ....(3.1,3)

° -k -k

"l • ^2

and a first approximation to the error is obtained. With 3 values of n we can similarly calculate £"0 ^^^ ^ ''^"^^ obtain a next approximation, and so on,

The method Just described occurred to the writer after

noticing that the error in the stop by step solution of a differential equation given by the process of Euler (of index unity) was almost exactly

halved when the number of intervals was doubled. After a general account 3 of the method had been prepared and circulated to the Aeronautical Research Council it was pointed out by R.A. Fairthorne

that the same method had been proposed earlier by L. P. Richardson ^>-^

who gave it as an example of what he called " the deferred approach to the liirdt". The practical value of the method, which is applicable to non-linear equations, has been demonstrated by a number of examples. It may be noted that the index varies from 1 in the original and

relatively crude process given by Euler in the eighteenth century to 4 in the process of Runge and Kutta °, The various methods arc briefly reviewed in the writer's paper 5,

numbers of steps allows the error to be assessed and partially corrected. If need be, a still larger number of intervals must finally be used.

3.2 Extension of the Method to Partial Differential Equations. The- method just described can sometimes be applied to partial differential problems. It is essential that a perfectly regular process be used and that an index should exist. When this is so, the error can be assessed as before when the results for two similar lattices are knovm. It may even be possible to dispense with this condition of similarity,

4. ESTIIvIATION OF THE ERRORS IN APFROIJMI'jn: SOLUTIONS OF LII'IEim PROBLEMS M-LEN AN APPROXIMATION TO THE GREEN'S FUNCTION

IS

m&m.

We shall suppose that we have aii' approximation if' to the solution of (2.3,2) which satisfies the linear and homogeneous boundary conditions exactly and gives a residual

( = f - A ( afa + /3 )

Then, if there is an approximation G to the Green's function appropriate to the homogeneous boundary conditions, we may derive an approximation to the error of deficiency in '"\4^^ at any point by forming the corresponding integral of the product x: G.^ throughout the region.

LIST OF REFERENCES Author DUNCAN,V\r. J. DUNCAN, W.J. DUNCAJ^r, W.J. RICHARDSON, L.P. RICHARDSON, L.P, RUNGE, C. Title etc.,

On the Torsion of Cylinders of Symmetrical Section.

Proc.Roy.Soc.,A, ¥01.136, p.95. 1932. Torsion and Flexure of Cylinders

and Tubes.

R. & M. 1444, Feb..1932. Technique of the Step by Step

Integration of Ordinary Differential Equations,

Report No.4 of the College cf Aeronautics. Feb.1947.

(To appear in the Phil.Mag,). The Approximate Arithmetical

Solution by Finite Differences of physical Problems involving Differential Equations, with an Application to the Stresses in

a Masonry Dam,

Phil.Trans. Roy.Soc.,A, Vol.210, p.307. 1911.

How to Solve Differential Equations Approximately by /irithmetic. Math.Gazette, Vol. XII, No.177, p.415, 1925.

Graphical Methods.

Columbia university Press, New York. 1912,